Logarithmic curve

Most pectin solutions behave like Newtonian liquids below a pectin concentration of about 1 % (w/w). Onogi (1966) derived the critical concentration of polymer solutions from plotting the double logarithmic curves of viscosity (ii) against concentration at constant shear rates. Each curve consists of two straight lines intersecting at the critical concentration. At higher... 

When the rotations are plotted as ordinates against the times as abscissae, the observed points can be connected by a logarithmic curve, which is flatter in the later stages of the reaction and indicates the order as unimolecular. The abscissa of the point in this curve corresponding to a rotation of 0° gives the zero-rotation time , which is to some extent a measure of the activity of the enzyme solution employed. 

A small addition to a logarithmic curve will be very evident at the lower end of the curve and virtually invisible at the top. For good matches with experimental curves plotted on a logarithmic scale (to see small peaks) it is necessary to add a background noise. With a well set up system, this is essentially the detector background, which may be measured at an angle remote from the substrate peak. 

Zone III the E vs ln(l — j/ji ) logarithmic curve corresponds to concentration polarization, which results from the limiting value ji of the mass transfer limiting current density for the reactive species and reaction products to and/or from the electrode active sites an increase inji from 1.4 to 2.2 Acm leads to a further... 

Figure 3.82 illustrates this with a logarithmic curve ending with an asymptote. 

Hexagonal and trigonal crystals. A% chart for these crystals can be constructed by plotting values of og(h2J -hk- -k2) along MA and values of log l2 along NC, and joining them by logarithmic curves. The axial ratio... 

Fig. 2.3 Experimental current-time curve (a) and logarithmic curves (b) for the application of a constant potential to a graphite disc electrode of radius 0.5 mm (planar electrode) for the reduction of Fe(CN)g. ...

The floating material and the effluent DM were measured only after the eighth separation. The curve for DM in the effluent was inferred by assuming that 70% of DM in the effluent was released after the first separation (see experiment 3 for a justification) and by assuming that the release followed a logarithmic curve. The curve for floating material was obtained by mass balance. The suspended stover recovered after the first separation was considered to be part of the floating material. 

By changing the angle (really distance or "travel" time) between the sample and the chamber, counts at different intensities were registered. Plots of activity versus angle gave the usual type of semi-logarithmic curve from which the half-life could be determined (down to about 20 mseconds). 

The values for —10° and —40° C. were obtained by interpolation, with the help of the logarithmic curves shown in Figure 6. 

Returning to the formula above (No. 2) it will be seen that the value of x (that the concentration of the solid in the solution) varies with time according to f logarithmic curve. Fig. 1. We would naturally suppose that the speed of dissoluti would decrease as the solvent becomes more saturated with the solute but it in interest to note the curve that relates to these two factors. 

For this study, we selected 42 representative FGs from a list of more than 500 substituents compiled by Hansch et a/.128 The electrophilicity index co for the whole series of ethylene derivatives, together with the experimental electronic substituent effects from the reactivity index co was as follows we first compared the computed co values with the experimental crp (see Figure 8). The analysis revealed a poor linear correlation between both quantities, with a regression coefficient R2 = 0.53. A better correlation was found when all the 42 points were fitted to a logarithm curve (.R2 = 0.84, see Figure 8). 

The rate of evaporation is very rapid immediately after a spill and then slows considerably. About 80% of evaporation occurs in the first few days after a spill, which can be seen in Figure 6. The evaporation of most oils follows a logarithmic curve with time. Some oils such as diesel fuel, however, evaporate as the square root of time, at least for the first few days. This means that the evaporation rate slows very rapidly in both cases. 

Tbe luminescence of GGA solutions were estimated using the bacterial chip system in order to produce a calibration plot for measurement of pollution containing biodegradable substances. The luminescent intensity was correlated with concentration of BOD standard (GGA) solution. The bioluminescence increased linearly with concentration up to approximately 50 ppm. Measurement of BOD values less than 50 ppm has been achieved. The bght intensity reached saturation at concentrations of over 100 ppm. Detection limit was approximately 1(X) ppm, and the minimum measurable BOD was 1 ppm. Based on the logarithmic curve which was obtained by the measurement with a chemi-imager, two linear calibration curves and equations were estimated and approximated... 

A series of standard solutions is prepared by successive dilutions of a stock solution. To each each solution, an excess but constant volume of the recommended ionic buffer (ISAB or TISAB) is added. Sample solutions are prepared in the same way. Next, for each of the standards, the PD across the electrodes is measured and a semi-logarithmic curve E = f(C ) is plotted (Figure 19.6). Finally, from this curve and the measured PD values for each of the sample solutions, obtained in the same manner, the concentration of the species i is obtained. The precision of measurements can be deduced from equation 19.5 ... 

In brief, it can be said that in the attractive domain, the Flory-Huggins theory describes the demixtion curve in a reasonable way. However, the renormalization theory of critical phenomena shows that the top of the demixtion curve is not parabolic but much flatter. Moreover, the tricritical theory shows that when cp increases, there is no oblique asymptote but a limiting logarithmic curve (see Section 6). Actually, experiments seem to confirm these more recent theories. Nevertheless, the Flory-Huggins theory remains a very interesting approximation whose continuous limit will now be studied. 

Figure I. Logarithmic curves of creep compliance vs. time for TCP Granite...

Similarly, the group of logarithmic curves of cohesion vs. time of the rock are obtained shown in Figure 3, by a series of triaxial compressive creep tests at seven temperatures from 20°C to 300°C. In the same way as mentioned above we can also obtain the main curve (Figure 4) at the reference temperature at 20°C as well as the corresponding shift factor parameters (Table 2). 

Figure 3. Logarithmic curves of cohesion vs. time at different temperatures...

The accumulation of ammonia in the blood of a.nimals that have been operated upon follows a very regular course, and corresponds practically to a logarithmic curve whose constant, calculated by assuming the quantity of active substance fixed, is well verified by experiment. The sudden arrest in the... 

Retardation spectra Lp were determined from the Jp(t) curves of Figure 5.10. The results reduced to Tg = -20 C are shown in Figure 5.8, where the principal maximum concentration of retardation mechanisms is shown to be at correspondingly longer times as Tg is increased with additional crosslinking. The logarithmic curves at shorter reduced times showing a slope of 1/3 for all three fluoroelastomers are consequences of the assumed Andrade creep (Plazek, 1960 Plazek and O Rourke, 1971), J(t) = Jg + at short times used to determine Jg. 

Still, there is an important difference here. A vacuum plays the role of solvent A as a medium in which the substances are dispersed. While A continues to exist even at finite pressures when no other substances are distributed within it, this is not true for a vacuum. Starting from a state of very low total pressure po, we extrapolate from pb = Xb Po to Pb = Xb p at constant temperature T and obtain a value which is used as basic value p 3 of fh chemical potential pb- Extrapolation is done along the ideal logarithmic curve p = + RTln p/p ). Differently from the basic... 

Figure 4.5. Double logarithmic curve / , us 0T for the system />-polystyrene- -//-polystyrene-hcyclohcxaiie. I he overall polymer concentration i.s c = 15 10 g/cm. The horizontal line defines R, in the 0 conditions (Cotton et al., 1976) (Reprinted with permiseion from J.P.Cotton, M.Nierlich, K.Bone, M.Daoud, B.Karnoux, G.Jannink, R.Dupplesix, C.Picot. J. Chein. Phye. 65 (1976) 1101-1108. Copyright 1976 American Institute of Physii s)...

Schematic representation of (transient) equilibrium. The half-lif parent nuciide considered is 10 units, while that of the daughter is 1 unit. Note that the maximum of the daughter s activity (seeO Eq. (7.58)) is reached somewhat later than that of the total activity. Note also thatO Eq. (7.53) works quite well, i.e., the logarithmic curves of 7li and k-2, are already practically parallel at t 6.6, which means that the ratio S constant...

---